Debt and Cost of Capital





Kerry Back

Overview

  • Adding debt increases enterprise value because of the tax savings, up to a point.
  • Enterprise value is free cash flow discounted at the WACC.
    • Adding debt does not change FCF.
    • So adding debt must reduce the WACC.
  • The effect is through taxes, not just because debt is cheaper capital.

Example

  • All equity firm
  • Issues perpetual risk-free debt with interest payment \(X\)
  • Value of debt is \(D=X/r_f\).
  • Tax savings per year is \(tX\), where \(t=\) tax rate.
  • Government’s claim is reduced by \(tX/r_f = tD\).
  • Enterprise value is increased by \(tD\).

If there were no taxes …

  • Enterprise value \(EV\) is unchanged
  • New equity value is \(E = EV-D\).
  • All risk is now borne by \(E\) dollars of capital rather than \(EV\), so risk per dollar increases by \(EV/E\).
  • Risk premium of equity should increase by \(EV/E\).
  • New cost of equity is

\[r_{\text{new}} = r_f + (EV/E) (r_{\text{old}}-r_f)\]

  • Without taxes, WACC is

\[\frac{E}{EV} \left[r_f + \frac{EV}{E}(r_{\text{old}}-r_f)\right] + \left(1-\frac{E}{EV}\right)r_f\]

  • This is \(r_{\text{old}}\).
  • Leverage wouldn’t change cost of capital if there were no taxes.

But there are taxes …

  • With taxes, leverage still increases the cost of equity.
  • With risk-free debt as before, all risk is borne by \(E\) dollars of capital rather than the old enterprise value.
  • Old enterprise value is the new enterprise value \(EV\) minus \(tD\).

\[EV - tD = E + D - tD = E + (1-t)D\]

  • So risk per dollar of capital rises by a factor of

\[\frac{E+(1-t)D}{E}= 1 + (1-t)\frac{D}{E}\]

  • So equity risk rises less when there are taxes.
  • The reason it rises less is that the IRS shares the risk increase.
  • Because equity risk rises less, the WACC falls (straightforward algebra).
  • Leverage reduces the cost of capital - up to a point.